Physics Notes - Herong's Tutorial Notes

∟Phase Space and Phase Portrait

∟Motion Equations of Linear Systems

This section provides a quick introduction of motion equations of linear systems, which are first order linear differential equations of the canonical coordinates.

What Is Linear System? A Linear System is a system where motion equations are first order linear differential equations of the canonical coordinates. In this case, motion equations can be expressed in the vector form as shown below: 

x' = Ax + b                            (P.12)
where: 
  x is the vector of [q1, q2, ..., p1, p2, ....]
  x' is the vector of dx/dt 
  A is a matrix of coefficients
  b is a constant vector

For a Linear System of a single object with 1 degree of freedom, vector x is [q, p] and x' is [q', p']. The motion equations can be expressed as: 

  |q'| = |a11, a12| . |q| + |b1|  
  |p'|   |a21, a22|   |p|   |b2|

Now let's look at those systems of motions presented earlier:

1. The Free Fall Motion of a single object is a linear system. The motion equations can be expressed as: 

  m*g = -p'                             (P.4)
  p/m = q'                              (P.5)

In vector form:
  |q'| = |0, 1/m| . |q| + |   0|  
  |p'|   |0,   0|   |p|   |-m*g|

2. An object on a spring moving horizontally is a linear system. The motion equations can be expressed as: 

  kq = -p'                              (P.7)
  p/m = q'                              (P.8)

In vector form:
  |q'| = | 0, 1/m| . |q| + |0|  
  |p'|   |-k,   0|   |p|   |0|

3. A simple pendulum is not a linear system, because p' is not linearly related to q: 

  m*g*l*sin(q) = -p'                   (P.10)
  p/(m*l*l) = q'                       (P.11)

Table of Contents

 About This Book

 Introduction of Space

 Introduction of Frame of Reference

 Introduction of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 Introduction of Hamiltonian

 Introduction of Lagrangian

 Introduction of Generalized Coordinates

►Phase Space and Phase Portrait

 What Is Phase Space

 What Is Phase Portrait

 Phase Portrait of Simple Harmonic Motion

 Phase Portrait of Pendulum Motion

►Motion Equations of Linear Systems

 Phase Portraits of 2-D Homogeneous Linear Systems

 References

 Full Version in PDF/ePUB